## Exponential Families

The exponential family of distributions are a particularly tractable, yet broad, class of probability distributions. They are tractable because of a particularly nice [Fenchel] duality relationship between natural parameters and moment parameters. Moment parameters can be estimated by taking the empirical mean of sufficient statistics and the duality relationship can then recover an estimate of the distributions natural parameters.

Kalman filtering (and filtering in general) considers the following setting: we have a sequence of states $x_t$, which evolves under random perturbations over time. Unfortunately we cannot observe $x_t$, we can only observe some noisy function of $x_t$, namely, $y_t$. Our task is to find the best estimate of $x_t$ given our observations of $y_t$. Continue reading “Kalman Filter”
for $n=1,2,...$ where $\epsilon_n$ are unobservable random errors and $\beta_1,...,\beta_p$ are unknown parameters.
Typically for a regression problem, it is assumed that inputs $x_{1},...,x_{n}$ are given and errors are IID random variables. However, we now want to consider a setting where we sequentially choose inputs $x_i$ and then get observations $y_i$, and errors $\epsilon_i$ are a martingale difference sequence with respect to the filtration $\mathcal F_i$ generated by $\{ x_j, y_{j-1} : j\leq i \}$.